Working with Factors and Divisibility

factor game

One of the many GMAT math concepts that we learn in grammar school but probably don’t use very often in real life is how to work with factors. Factors are numbers that go into other numbers evenly. For example, 3 and 4 are factors of 12. Prime factors (numbers divisible only by one and themselves) help us realize whether one very large number is divisible by another.

For example, the prime factors of 108 are 2, 2, 3, 3, and 3 and the prime factors of 36 are 2, 2, 3 and 3. If we write those out as a fraction, we get:

2 × 2 × 3 × 3 × 3
    2 × 2 × 3 × 3

Every prime number in the denominator can be cancelled out by a prime number in the numerator, so 36 is a factor of 108. The numerator can have a lot of different prime factors left over. All that matters is that the prime factors in the denominator are cancelled out. Here’s another example with  18036:

2 × 2 × 3 × 3 × 5
    2 × 2 × 3 × 3

The 5 in the numerator doesn’t correspond to any numbers in the denominator, but that doesn’t matter. Every prime factor in the denominator can be cancelled out. For a counter example, let’s try 27036:

2 × 3 × 3 × 3 × 5
    2 × 2 × 3 × 3

There are two 2s in the denominator, but only one 2 in the numerator, so we are unable to cancel out all prime factors in the denominator, and we then know that 270 is not evenly divisible by 36.

So how can we use this concept on the GMAT? Let’s look at question 116 on page 168 in the GMAT Official Guide, 13th Edition. P, the numerator, will include every prime factor for every number 1 through 30 inclusive. However, the only prime number in the denominator is 3. So the question is really asking how many 3s there are in the numerator. However many 3s are in the numerator will be the same amount that can be in the denominator and be cancelled out to make 3k a factor of P. Take every number in the range that is a multiple of 3 (3, 6, 8, 12, 15, 18, 21, 24, 27, and 30) and count up the 3s. Remember that some may have more than 1 (for example, 27 = 3 × 3 × 3). The total is 14, answer C.

Let’s also have a look at question 77 on page 163. When we work with factors and multiplication, the prime factors in the denominator only have to be cancelled out by one prime factor in the numerator. But addition and subtraction are a bit more complicated in that every number in the numerator needs to have the same prime factors as the ones you want to cancel out in the denominator. Here is an example with (72 + 24)12:

(2 × 2 × 2 × 3 × 3) + (2 × 2 × 2 × 3)
                      2 × 2 × 3

The 2, 2, and 3 in the denominator exist in both the prime factors of 72 and 24.

(2 × 2 × 2 × 3 × 3) + (2 × 2 × 2 × 3)
                     2 × 2 × 3

So 12 is a factor of 72 + 24. However, if we try (46 + 24)12 we get

(2 × 23) + (2 × 2 × 3 × 3)
                2 × 3 × 3

We don’t have the right prime factors in 46 to be able to cancel out those in the denominator, so 12 is not a factor of 46 + 24. Since question 77 has addition, we know that the correct answer must have the same prime factors as both 20! and 17. Since 17 is a prime number, it is only divisible by 1 and 17, so 15 and 19 cannot be factors of the expression.

Factors can be more complicated than just a single number. So stay tuned and in a few weeks we’ll discuss a few more problem solving questions that use factors.

 

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You Bought the Books, Now What?

Calendar Card

A few weeks ago I wrote about how to find motivation to study for the GMAT and how to approach studying with a positive attitude. I’m sure that you took my advice to heart and have decided to buckle down and get to work so that you can have your GMAT experience behind you by the time the summer arrives. So today I’d like to offer you some tips about creating your study schedule. If you’ve read my book, you know that creating (and sticking to!) a study schedule is an important component of studying for the GMAT. My book has 35 pages devoted to helping you do just that, including a lot of great advice about how long you should study each day and what to study each day. This post will serve as a bit of a reminder about some of those tips, as well as offer you additional guidance, by talking about some of the most common mistakes GMAT test-takers make when studying.

1)      Skipping over the basics. Some people want to dive right into realistic GMAT questions, and for some people, that’s fine. Others, however, can’t remember how to add fractions or what the formula is for the area of a triangle. If you are struggling with the basic concepts – brush up on them first. The GMAT Official Guide has lots of basic math information before the problem solving section. Make flashcards for formulas and properties of numbers that you don’t remember. You can’t expect to solve the 37 quantitative questions in the time allotted if you are struggling with the concepts. You also can’t focus on good test taking strategies, like backsolving and estimation, if you can’t grasp what the question is asking.

2)      Avoiding topics they think they know well. If you think back to middle school and remember that you were an algebra-whiz kid, you might think you should focus on statistics and sentence correction. However, middle school was a long time ago and solving algebra questions on the GMAT is a whole different experience. It’s just as important that you practice the topics you think you are good at as those you have a hard time with. Keep in mind that the test is computer adaptive. If you get the easy and medium algebra questions correct right away, the test is going to give you some real whoppers.

3)      Avoiding topics they don’t like. On the flip side, don’t avoid the topics you know you have a hard time with (ahem…reading comprehension!). Don’t make the mistake of thinking that reading comprehension is only about a third of the verbal, and that if you ace sentence correction and critical reasoning, you’ll get a great score. Chances are that you will make at least a few mistakes in sentence correction and critical reasoning, so you can’t rely on those topics only for a high verbal score. If you spend a bit of time with those boring reading comprehension passages (or whatever your nemesis is), you might find that you can get a few extra points more easily than you think.

4)      Thinking more and more and more practice is the best approach. The study schedule that my book lays out includes a lot of time spent reviewing questions that you miss in your practice. Looking at the right answer and moving on immediately doesn’t help. Reviewing questions you missed means checking to see whether you made simple calculation errors, checking to see whether you missed an opportunity to use a strategy, and noting patterns in the types of questions you miss so you know what to focus your study efforts on. Use the “standardized” part of this standardized test to your advantage. Questions are about concepts and patterns, not individual questions. If you take the GMAT 20 times, you will see roughly the same 37 quantitative questions each time, just with different words and numbers. You don’t need to see an example of every type of question that has ever been on the GMAT. You just need to recognize the concept that is tested in each question and have an approach for that concept. Reviewing missed questions helps you figure those concepts out.

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How to Match the Answers to the Text in Reading Comp

Two parallel stacks of books on blue background

As you are probably used to hearing me say by now, every reading comprehension answer has to be clearly supported by information in the text. Even when a question is written as a hypothetical scenario, there will be evidence in the passage that supports the correct answer. However, it can be easy to let your mind wander outside of the information in the passage when a question is hypothetical. In today’s post, we’ll practice with two such questions so that we can master good techniques.

The first is question 112 on page 408 of the GMAT Official Guide, 13th Edition. The first step to answering this question correctly is taking the time to articulate the method that the author finds problematic. This is found in lines 7 – 9, which say that we shouldn’t remove a species that might be the keystone species then observe changes in the ecosystem. Now that we have identified the problem, we just have to look through the answers to find an experiment that does just that. Answer C explicitly mentions removing a species and then observing the ecosystem after that species is gone. This is correct.

While the answer to this question may have been relatively easy to identify, you can still look at the other answers to understand the ways in which the GMAT test writers attempt to trap you. Most of the wrong answers discuss topics related to keystone species that are discussed in the passage as well. Answers D and E discuss the influence of different environments on the keystone species, as is discussed in lines 118 – 122. Answer B discusses the possibility of another species occupying the keystone role, as is discussed in lines 27 – 29. On more difficult questions, if you don’t take the time to articulate the author’s position first, you could easily be attracted to one of these distractor answers.

Let’s try one that is more difficult – number 123 on page 413. Here, we need to first articulate the social constructivists’ version of technological determinism. The evidence is tricky to find because of the wording of the paragraph in which we find the answer (lines 25 – 32). You may need to read this paragraph carefully to understand that it does actually give the social constructivists’ version, not the true beliefs of the technological determinists. The constructivists’ version is that machinery imposes forms, and technology directly influences skills and work organization. Answer A is a direct match for that idea. Other answers, like B and C, support Clark, who refutes this version of technological determinism.

If you struggle with this question, remember to wary of answers that are extreme. Answer B uses the word all and answer D uses the word most. Such words are rarely supported by the passage. If you don’t see those extreme ideas expressed in the passage, they aren’t correct.

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How to Avoid More Data Sufficiency Traps

All About Math

Several months ago I wrote a post about not jumping to conclusions on data sufficiency questions. Sometimes, you will need to do more math on these questions than you think is necessary in order to avoid selecting a trap answer. Today’s post will be a continuation of the same idea, but we are going to focus on one particular trap – the too obvious answer.

Let’s start with question 163 on page 290 of the Official Guide, 13th Edition. Here, you might be tempted to select answer A and move on, since all A requires is that you divide 60 by 4. But when one statement is so obvious, you should take a long, hard look at the other statement. Since the first statement has tricked your mind into thinking this is a very easy question, your instinct will most likely be to gloss over the second statement because you already think you have the answer. However, if we look carefully at statement 2, we realize that we can create an equation with one variable, which means the statement is sufficient. If x is the original number of members, we divide 60 by x to get the value of the contribution of each member and then add $2 for the increase when (equals sign goes here) we take away 5 from the original number members, x. So, the equation is 60x + 2 = 60(x – 5).

Do you want to see a few more questions that try to trap you this way? Flip to page 288 and look at number 142. Statement 1 is obviously sufficient because it gives the total number of tosses, but statement two seems to be irrelevant because it discusses points earned based on results of the tosses. But here again we can create an equation with one variable. Heads, represented by x + 4 (according to the question stem) will be multiplied by 3 and tails, represented by x, will be multiplied by 1. So, the question is 3(x + 4) + 1(x) = 52. We then discover that x = 10, so there were 10 tails and 14 heads, for a total of 52 tosses.

One more? OK, let’s turn to page 284 and look at number 107. Statement 1 is sufficient because you would only need to divide the total number of frames by the seconds and then multiply by 60 to convert to minutes. But what about statement 2? Again, we can write a single variable equation: 6x + x = 14. This gives a value of 2 for x and a total of 12 minutes to run the cartoon. Although the information provided in statement two was entirely different from that in statement one, it was still sufficient to answer the question. Also, notice that 12 minutes is 720 seconds, which means that there are 24 frames per second. This is consistent with the information in statement 2. If the correct answer is D, the information in the two statements cannot contradict each other.

I’m not saying that every time one statement is too obvious, the other statement will always be sufficient and the correct answer will be D, but I am saying, once again, that you shouldn’t jump to conclusion. Be sure to give each statement its fair amount of attention so that you don’t miss an easy point.

 

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How To Reduce Stress on Test Day

STICC Lab Instruction Room

Of all the stresses that are involved in taking the GMAT, worrying about conditions at the test center should not be one of them. There are many potential issues that you can overcome or prepare for long before big day comes. Here are a few of them.

                       Noise: The test center is not a completely quiet environment. People in your center will be taking a variety of tests that start and end at different times, and have breaks at different times. You can expect people to be walking back and forth behind you, and you can expect a little chatter from outside as people enter and leave the room. Keep this in mind as you practice; don’t practice in a completely isolated and quiet place. You will be given headphones to help block the noise, but some people don’t find them very comfortable.

                       White Boards: You will be given white boards and a marker for your scratch work. Most people don’t find this problematic, but some test takers really have an affinity for the old number two pencil. If that sounds like you, pick up a felt tip pen and some white boards to practice on.

                       Missing Time During Breaks: You have two, optional breaks during the course of the test. However, when you enter and leave the test room, you need to check in and check out with the staff. If your test center is large or extremely busy, you may end up having to wait a few minutes until you can check back in. But if your break time is finished, your test will resume regardless of whether you are sitting in front of your computer. Be aware of the procedures and the volume of people in your test center and don’t plan on checking back in from break when there are just five seconds remaining.

                       Hunger and Cold: Dress in layers and bring a snack to eat during your break. There are lockers for your personal items just outside the test room. However, you should know that you aren’t allowed to use a cell phone during your break, so you should just leave it in your car to avoid the temptation and possibly have your score invalidated.  

                       Getting Lost: Know where your test center is beforehand and plan to arrive 30 minutes early. You will forfeit your appointment and test fee if you are 15 minutes late. Every location is different, so visiting the center before test day can do a lot to help calm your nerves. You can also check out the forums to find out about other people’s experiences at your center so that you won’t be surprised by the conditions. You can even sign up to take a practice test in your test center if your nerves just seem to be unshakeable.

 But even the most prepared test takers run into unexpected circumstances. Some people have reported computer glitches that erased their scores part way through the test, cleaning crews entering during test times, power outages (although all test centers should have generators for backup), or inclement weather that prevented them from getting to the test center. In the case of extreme weather (hurricanes, for example) the test center will be shut down and all test takers scheduled that day will be able to reschedule free of charge (this possibility should be good motivation not to wait to the the GMAT until the day before your B-school application is due!). However, be aware that if something happens in the center in the middle of your test, you will have to take the GMAT over from the beginning; it isn’t possible to resume an interrupted test. In the event that something horrible, such as a computer glitch, should happen during your testing experience, your options for recourse are limited, but it never hurts to try. You can, and should, file a complaint. GMAC will investigate and in the case that your complaint is found valid, you will be able to retest at no fee.

Knowing what to expect can go a long way toward calming your nerves. GMAT preparation is not just about knowing the content; it’s about being familiar with the entire process.

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Fill in the Gap

Assumption, Minnesota

Assumption questions can be quite tricky on the GMAT because the answer choices are often answers to other critical reasoning question types, such as inference questions and weaken questions. The arguments that are given for assumption questions include the premises (the facts) and the conclusion (what the author wants you to believe). The assumption is a statement that must also be true in order for the stated premises to lead to the conclusion. It is missing information, the link between the premise(s) and conclusion. So to find the assumption, you have to think about what the written argument is lacking. Generally, you can do this just by focusing on the key words in the premises and conclusions. The correct answers to assumption questions never contain unrelated information.

Let’s try question number 75 on page 522 of the GMAT Official Guide, 13th Edition. The first sentence is the premise and the second is the conclusion, which is indicated by the word therefore. The premise focuses on the transfer of information from lower levels to the superior. The conclusion focuses on the same. Since the assumption needs to link those two sentences, the answer should be limited to that concept as well.  Here are the options:

                (A) The word should indicates another conclusion, not an assumption.

                (B) The word should indicates another conclusion, not an assumption.

                (C) Problem-solving ability is off-topic.

                (D) Yes – this answer has the same focus as the premise and conclusion. We cannot conclude that the chief executive is less well informed unless we know that the subordinates who have the problem are the only ones who initiate the flow of information. If employees at higher levels have direct access to the information, it will not be as distorted when it reaches the top.

                (E) This answer weakens the argument because it says that some employees won’t distort the information.

 

Number 83 on page 525 is another, slightly harder, example of an assumption question. Here we have two premises: (1) some people are allergic to sulfites in wine and (2) some winemakers don’t add sulfites to their wine. The conclusion is that the people who are allergic can drink the wines that don’t have the added sulfites. Again, remember that the assumption is a link from premises to conclusion, so you need to find an answer that mentions only sulfites, wine, and allergies.

                (A) Preservative effect is off-topic.

                (B) The form of the sulfite doesn’t matter because the argument specifies that none is added at all.

                (C) Other beverages are irrelevant.

                (D) Other substances are irrelevant.

                (E) Yes – this answer includes all of the key subjects from the premises and conclusion and indicates another way that people might have an allergic reaction from sulfites in wine. The possibility of the sulfites occurring naturally, since the argument only mentions added sulfites, must be addressed for the conclusion to be valid.

 

 Let’s try one more – a really tricky one this time: question 93 on page 529. The first sentence is the premise and the concept discussed is the retaliatory closing of trade markets. The conclusion, phrased as a hypothetical statement in this case, is that retaliation would lead to the end of trade. So, the correct assumption obviously needs to discuss closed markets and the impact on trade.

                (A) This answer somewhat weakens the argument, or at least implies that we need not worry about the commentator’s hypothetical scenario.

                (B) The word should indicates another conclusion, not an assumption.

                (C) The word should indicates another conclusion, not an assumption.

                (D) Yes – this answer discusses closed markets and explains that a country can follow the theory of trade retaliation but will still continue to trade because its trade partners have not closed any of their markets. Because the commentator’s conclusion is so extreme—no country would trade with anotherhe is assuming that there is always some market that is closed between any two countries.

                (E) Foreigners and domestic products are off-topic.

So remember, for assumption questions, focus on the key words or concepts in the premises and conclusion, and look for an answer that specifically address those, without bringing in any new information.

 

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Percents: More than Calculating Tips

20% sign

Even if you have already learned how to accurately and quickly calculate a tip using mental math, you may still be struggling with percentage questions on the GMAT. Today we will look at three questions that test percents in different ways.

 

Let’s start with problem 71 on page 162 of the Official Guide, 13th Edition. This question is a very straightforward percent question as far as setting up the equation. The total is 2,420 stocks which has to be broken down into two categories, one of which is 20 percent greater than the other. Therefore, the equation is: 2,420 = x + 1.2x

Solving for x results in 1,100, which, not coincidentally, is answer choice C. But C is NOT correct! The trick in a question such as this which seems to be testing reallybasic concepts about percents is in the wording.  With all the language changes from different, higher, lower, percent greater you can quite easily become confused and answer the question incorrectly, not because you made a math mistake but simply because you misread it. Always double check the question. In this case, because the question is looking for the number of stocks that closed higher (represented in our original equation as 1.2x) , we should multiply 1,100 by 1.2 to find answer D, 1,320.

 

Now let’s move on and look at a different type of percent problem. Flip to page 164 and look at question 84. The Official Guide gives a very official explanation for this question with yet another formula you might think you have to master to do well on the GMAT. Instead, let’s think about what compounding does. Compounding means that at some point during the year (every three months if compounded quarterly, for example) the interest already earned is added to the principal and you start earning interest on that interest as well. Since the balance has been increased, by the end of the year, you will earn a little bit more than if you just had simple annual interest.  

Therefore, even if you don’t know the formula, you can use process of elimination. The simple interest on $10,000 at a rate of 8 percent would be $800. Knowing that compounded interest results in a higher amount, you can eliminate answers D and E. You should also be able to tell that answer A is too large. Answer C ($816) is correct. Half the interest ($400) is earned by the 6 month, which is also the compounding point in this question, so you will also earn the remaining 4 percent not only on the original $10,000 but also on that $400. That comes out to $16 more than the simple $800 earned for the year.

 

A third way that percents are tested is with percent change. Percent change has a simple formula, way simpler than the Official Guide would like you to believe. That formula is: differenceoriginal

The explanation for question 114 on page 168 uses way more variables than are necessary. All you need to use is a single x. If x represents the original number of workers, then your formula becomes (.16)x – (.09)(1.2x)(.16)x. The numerator is the difference (16 percent originally minus the 9 percent new value in an overall larger population that is 20 percent greater), and the denominator is the original 16 percent. The result is .325, which is the same as 32.5 percent. Since the question specifies that you are looking for an approximate percent change, you should feel confident selecting answer B, 30 percent.

 

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But I Don’t Want To Study for the GMAT!

071/365 Crying

 

Spring is upon us…well, upon some of us anyways. But wherever you live, you are probably feeling the desire to step out from winter hibernation and enjoy the fresh air. Unfortunately, we are moving into prime season for B-School applications and that means studying for the GMAT. There’s no escaping the GMAT requirement for most schools and you certainly don’t want to take the test twice, so it’s best to buckle down for the next 30 days, study hard, and get it out of the way. And studying for this test doesn’t have to be all bad. Here are a few ways that you can turn your GMAT studying experience into a beneficial one.

1)      The median age for business school students is about 27. If you are one of these typical students, you have probably been out of college for about five years and that means you are out of the habit of studying. Getting back into a good academic routine can be more difficult than you think, especially if you plan to join a PMBA or EMBA program and work while you study, or if you have a family. Use your GMAT preparation experience to establish good habits now. Set aside a specific time of the day to study and commit to turning off your phone and email during that time. Let the people you live with know that you should be disturbed during that time, or, even better, go to the local library or a coffee house.

 2)      Along with the first point, another consequence of having been out of school for awhile is that you may have forgotten many of the basic quantitative skills you need to succeed in B-School. The GMAT gives you the chance to refresh those skills before you are in a classroom environment where the pressure will be on. Don’t gloss over all the fundamental math concept review in the Official Guide; read it carefully and master it. This is especially important for applicants who do not have an undergraduate background in business or math and might be required to take some fundamental quantitative courses before they are granted full admission to an MBA program.

 3)      Use your GMAT preparation as an opportunity to begin networking. If you take a class, you will meet other classmates that you may even end up in school with. If you study on your own, you can join a study group or participate in online forums. That way, you have other people to ask questions when you don’t understand GMAT problems, review application essays with, and possibly connect with in useful ways later on in your career.

 4)      Embrace the competitive spirit! B-School isn’t going to be easy. There are scholarship applications, case competitions, and many opportunities to show how you can shine above your classmates. Look at the GMAT as that first challenge. Rather than viewing it as a cumbersome obstacle, view it as a way to demonstrate your mental prowess. Then you will be in the right frame of mind to impress your professors, get great recommendations, and land an amazing job.

 Remember that the GMAT is just one step in obtaining your MBA, but it can be a very important step. Make the most of your study time and you will see very real benefit come from it. And don’t worry about missing out on that lovely spring breeze – you can always study outside!

 

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Don’t Repeat or Say Again what You’ve Already Said

The Grammar Bible

 

You don’t have to read the grammar bible to do well on GMAT sentence correction. Most questions test the same small handful of grammar rules such as parallelism, agreement, verb form, and logical predication. But every now and then, the test writers will throw a little something new in the mix. If you’ve been working through the Official Guide 13th Edition these curveballs won’t be entirely new for you, but it does take a keen eye to notice them sometimes. In today’s post, we are going to look at two of those topics that you may not normally be on the lookout for: redundancy and diction.

On page 681, there are two questions that deal with redundancy. You might think at first that question 55 is completely fine. It certainly sounds fine and there is a solid reason to eliminate some of the answers. Choices C and D both lack agreement between programs and enables. But what about the other options? Choice B eliminates be able. Why would an answer do that? When a seemingly random word or phrase is eliminated, that’s when you might want to check for a redundancy problem. Earlier in the sentence we have the verb enable. Therefore, be able is redundant and should be eliminated.

Question 56 has the same problem but is a bit more complicated. Also is redundant with both, but that only eliminates choices A and B. How do you know if repeating the preposition of would also be redundant? Because of comes before the word both, you don’t need repeat it. Remember, when in doubt, select the more concise answer.

Now look at question 83 on page 687. If you are attentive to the grammar errors that the GMAT commonly tests, you’ll probably see the subject/verb agreement error first, dioxins…comes. That will get your choices down to B, D, and E. If you don’t see another problem with the sentence, you can look for differences among the answer choices. There is a clear division between much and many. This is an example of a diction error that isn’t tested very often, but is worth noting because it follows a rule that is very easy to learn. Much is for collective concepts or items that can’t be made plural (snow, disgust, intelligence) and many is used with items that can be counted individually (people, apples, countries). Dioxins is plural, so the word must be countable and requires the word many. Be sure to check out the Sentence Correction section in my book for more examples of commonly confused words that you might see on test day.

So what’s the take away from today’s post? No, you don’t have to take a remedial English class, but be sure as you work through the Official Guide questions that you take note of why the wrong answers are wrong and keep making notes about types of errors you see so that you can look for those on test day.

 

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To the Right, To the Right

Simple geometric proof of the pythagorean theorem

When working geometry problems on the GMAT, you are certain to come across several that deal with right triangles and your old middle school friend, Pythagoras.  Remember him? He was the a2 + b2 = c2 guy. While Pythagoras and his theorem won’t let you down, they will waste your time. Remember that you don’t have a calculator and sometimes squared numbers can get very large. You are also likely to make mistakes when working under the pressure of the test and you can’t go back to check your work once you move to the next question.

The good news is that right triangle problems often test one of several commonly seen special right triangles instead of arbitrary ones. If you have the dimensions of the special right triangles memorized and are on the lookout for them, you can save yourself some time and improve your accuracy. Let’s look at them.

Question 159 on page 174 of the Official Guide, 13th edition has a right triangle with a height of 5 and a hypotenuse of 13 (the distance from A to B). If you were aware of the a 5, 12, 13 is a special right triangle, you could instantly fill in the length of the base of the rudder and then split the shape into a rectangle and a right triangle to calculate the total area. The 5, 12, 13 right triangle is known as a Pythagorean triple. Another is the 3, 4, 5 right triangle. Be on the lookout for those (and their multiples, i.e. a 6, 8, 10) on your test.

Question 161 on the next page gives us an example of some other special right triangles. Here, the height is again 5, but the hypotenuse (the length from V to R) is 10, a relationship of x to 2x. Given those two measurements in a right triangle, the length of the third side will then have the relationship x√3. We can go even further and know automatically that the degree measures will be 30°, 60°, and 90°. Remember that the shortest side of a triangle is opposite the smallest angle and the longest side is opposite the biggest angle. So if you drew a line from V to R, the angle created by the height and hypotenuse would be 60° The roots in the answer choices should also be a clue to you that you may be dealing with one of these special right triangles, and in fact, the answer is A. The GMAT rarely requires you to solve out √3 (which you can estimate as 1.7) or √2 (which you can estimate as 1.4). Answer choice B has √2 in it, which is for a different special right triangle with the measurements of 45°, 45°, and 90° with sides in a relationship of x, x, x√2.

Remember, these special triangles won’t always be there, but having them in the back of your mind as a possibility can be a useful extra boost for your score.

 

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